LMFDB - Newform orbit 1767.1.eh.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1767,1,Mod(86,1767)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1767.86"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1767, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 85, 39])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1767 = 3 \cdot 19 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1767.eh (of order $90$, degree $24$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.881847877323$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{45})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{90}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{90} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + \zeta_{90}^{37} q^{3} - \zeta_{90}^{26} q^{4} + ( - \zeta_{90}^{38} - \zeta_{90}^{34}) q^{7} - \zeta_{90}^{29} q^{9} + \zeta_{90}^{18} q^{12} + ( - \zeta_{90}^{28} - \zeta_{90}^{6}) q^{13} - \zeta_{90}^{7} q^{16} + \cdots + ( - \zeta_{90}^{37} + \zeta_{90}^{25}) q^{97} +O(q^{100}) $ q + z^37 * q^3 - z^26 * q^4 + (-z^38 - z^34) * q^7 - z^29 * q^9 + z^18 * q^12 + (-z^28 - z^6) * q^13 - z^7 * q^16 + z^24 * q^19 + (z^30 + z^26) * q^21 - z^5 * q^25 + z^21 * q^27 + (-z^19 - z^15) * q^28 - z^2 * q^31 - z^10 * q^36 + (z^38 + z^22) * q^37 + (-z^43 + z^20) * q^39 + (-z^20 + z^6) * q^43 - z^44 * q^48 + (-z^31 - z^27 - z^23) * q^49 + (z^32 - z^9) * q^52 - z^16 * q^57 + (z^16 + z^9) * q^61 + (-z^22 - z^18) * q^63 + z^33 * q^64 + (-z^36 + z^4) * q^67 + (-z^33 + z^29) * q^73 - z^42 * q^75 + z^5 * q^76 + (-z^35 - z^3) * q^79 - z^13 * q^81 + (z^11 + z^7) * q^84 + (z^44 + z^40 - z^21 - z^17) * q^91 - z^39 * q^93 + (-z^37 + z^25) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{12} - 3 q^{13} + 3 q^{19} - 12 q^{21} - 3 q^{27} - 12 q^{28} + 3 q^{43} - 6 q^{49} - 6 q^{52} + 6 q^{61} + 6 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 3 q^{75} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) $ 24 * q - 6 * q^12 - 3 * q^13 + 3 * q^19 - 12 * q^21 - 3 * q^27 - 12 * q^28 + 3 * q^43 - 6 * q^49 - 6 * q^52 + 6 * q^61 + 6 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 3 * q^75 + 3 * q^79 + 3 * q^91 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1767\mathbb{Z}\right)^\times$.

$n$	$590$	$685$	$838$
$\chi(n)$	$-1$	$-\zeta_{90}^{24}$	$\zeta_{90}^{35}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

86.1

0.961262	−	0.275637i
−0.997564	+	0.0697565i
0.848048	+	0.529919i
0.438371	−	0.898794i
0.0348995	+	0.999391i
−0.882948	+	0.469472i
0.990268	−	0.139173i
0.438371	+	0.898794i
−0.719340	+	0.694658i
−0.374607	−	0.927184i
−0.374607	+	0.927184i
0.990268	+	0.139173i
−0.241922	+	0.970296i
−0.997564	−	0.0697565i
−0.719340	−	0.694658i
−0.241922	−	0.970296i
−0.882948	−	0.469472i
−0.615661	−	0.788011i
0.559193	+	0.829038i
0.0348995	−	0.999391i

0.615661

−

0.788011i

−0.559193

0.829038i

1.37217

−

0.996940i

−0.241922

−

0.970296i

110.1

−0.848048

−

0.529919i

0.241922

0.970296i

1.60229

1.16413i

0.438371

0.898794i

167.1

0.241922

−

0.970296i

0.374607

−

0.927184i

−0.270928

−

0.833831i

−0.882948

−

0.469472i

203.1

0.882948

−

0.469472i

0.719340

−

0.694658i

−0.996161

−

0.723753i

0.559193

−

0.829038i

230.1

−0.961262

−

0.275637i

0.615661

−

0.788011i

0.616528

−

1.89748i

0.848048

0.529919i

260.1

0.719340

0.694658i

−0.990268

0.139173i

−0.345600

−

1.06365i

0.0348995

0.999391i

269.1

−0.438371

−

0.898794i

0.882948

−

0.469472i

−0.594092

−

1.82843i

−0.615661

0.788011i

383.1

0.882948

0.469472i

0.719340

0.694658i

−0.996161

0.723753i

0.559193

0.829038i

611.1

−0.990268

0.139173i

−0.438371

0.898794i

0.0564686

0.0410268i

0.961262

−

0.275637i

623.1

0.997564

−

0.0697565i

−0.848048

0.529919i

0.444576

−

1.36827i

0.990268

−

0.139173i

641.1

0.997564

0.0697565i

−0.848048

−

0.529919i

0.444576

1.36827i

0.990268

0.139173i

716.1

−0.438371

0.898794i

0.882948

0.469472i

−0.594092

1.82843i

−0.615661

−

0.788011i

737.1

0.374607

0.927184i

0.997564

0.0697565i

−1.42864

1.03797i

−0.719340

0.694658i

755.1

−0.848048

0.529919i

0.241922

−

0.970296i

1.60229

−

1.16413i

0.438371

−

0.898794i

830.1

−0.990268

−

0.139173i

−0.438371

−

0.898794i

0.0564686

−

0.0410268i

0.961262

0.275637i

983.1

0.374607

−

0.927184i

0.997564

−

0.0697565i

−1.42864

−

1.03797i

−0.719340

−

0.694658i

1067.1

0.719340

−

0.694658i

−0.990268

−

0.139173i

−0.345600

1.06365i

0.0348995

−

0.999391i

1199.1

−0.559193

0.829038i

−0.0348995

0.999391i

0.149516

0.460163i

−0.374607

−

0.927184i

1226.1

−0.0348995

0.999391i

−0.961262

−

0.275637i

−0.606126

−

0.440376i

−0.997564

−

0.0697565i

1283.1

−0.961262

0.275637i

0.615661

0.788011i

0.616528

1.89748i

0.848048

−

0.529919i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
589.ce	even	90	1	inner
1767.eh	odd	90	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1767.1.eh.a	✓	24
3.b	odd	2	1	CM	1767.1.eh.a	✓	24
19.f	odd	18	1		1767.1.ew.a	yes	24
31.h	odd	30	1		1767.1.ew.a	yes	24
57.j	even	18	1		1767.1.ew.a	yes	24
93.p	even	30	1		1767.1.ew.a	yes	24
589.ce	even	90	1	inner	1767.1.eh.a	✓	24
1767.eh	odd	90	1	inner	1767.1.eh.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1767.1.eh.a	✓	24	1.a	even	1	1	trivial
1767.1.eh.a	✓	24	3.b	odd	2	1	CM
1767.1.eh.a	✓	24	589.ce	even	90	1	inner
1767.1.eh.a	✓	24	1767.eh	odd	90	1	inner
1767.1.ew.a	yes	24	19.f	odd	18	1
1767.1.ew.a	yes	24	31.h	odd	30	1
1767.1.ew.a	yes	24	57.j	even	18	1
1767.1.ew.a	yes	24	93.p	even	30	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1767, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} $ T^24
$3$	$ T^{24} + T^{21} + \cdots + 1 $ T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$5$	$ T^{24} $ T^24
$7$	$ T^{24} + 6 T^{22} + \cdots + 1 $ T^24 + 6T^22 + 2T^21 + 27T^20 + 18T^19 + 111T^18 + 108T^17 + 456T^16 + 404T^15 + 1062T^14 + 1230T^13 + 2592T^12 + 3204T^11 + 5589T^10 + 6560T^9 + 8397T^8 + 7020T^7 + 5315T^6 + 2556T^5 + 1167T^4 + 294T^3 + 162T^2 - 21T + 1
$11$	$ T^{24} $ T^24
$13$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 3T^22 - T^21 - 30T^20 - 75T^19 - 72T^18 - 15T^17 + 222T^16 + 446T^15 + 363T^14 + 525T^13 + 516T^12 + 498T^11 + 579T^10 - 241T^9 + 1401T^8 + 2064T^7 + 1091T^6 + 330T^5 + 900T^4 + 444T^3 + 156T^2 - 18T + 1
$17$	$ T^{24} $ T^24
$19$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$23$	$ T^{24} $ T^24
$29$	$ T^{24} $ T^24
$31$	$ T^{24} + T^{21} + \cdots + 1 $ T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$37$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 - 2T^21 + 90T^20 - 21T^19 + 425T^18 - 135T^17 + 1476T^16 - 524T^15 + 3618T^14 - 1419T^13 + 6625T^12 - 2556T^11 + 8322T^10 - 3139T^9 + 7560T^8 - 2664T^7 + 4193T^6 - 1377T^5 + 1563T^4 - 370T^3 + 180T^2 + 12T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 - 2T^21 + 6T^20 - 21T^19 + 51T^18 - 54T^17 + 168T^16 - 510T^15 + 834T^14 - 762T^13 + 234T^12 - 702T^11 + 2535T^10 - 3311T^9 + 3948T^8 - 1812T^7 + 512T^6 + 384T^5 + 243T^4 + 298T^3 + 105T^2 + 15*T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} - 6 T^{23} + \cdots + 1 $ T^24 - 6T^23 + 21T^22 - 57T^21 + 135T^20 - 291T^19 + 541T^18 - 876T^17 + 1272T^16 - 1659T^15 + 1761T^14 - 696T^13 - 1720T^12 + 3921T^11 - 3483T^10 - 120T^9 + 2640T^8 - 1872T^7 + 392T^6 - 63T^5 + 462T^4 + 237T^3 + 3T^2 - 3*T + 1
$67$	$ T^{24} - 6 T^{23} + \cdots + 1 $ T^24 - 6T^23 + 21T^22 - 57T^21 + 135T^20 - 291T^19 + 541T^18 - 876T^17 + 1272T^16 - 1659T^15 + 1761T^14 - 696T^13 - 1720T^12 + 3921T^11 - 3483T^10 - 120T^9 + 2640T^8 - 1872T^7 + 392T^6 - 63T^5 + 462T^4 + 237T^3 + 3T^2 - 3*T + 1
$71$	$ T^{24} $ T^24
$73$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + 3T^21 - 24T^20 + 39T^19 + 6T^18 - 99T^17 + 183T^16 - 185T^15 + 129T^14 - 492T^13 + 994T^12 - 1872T^11 + 2325T^10 - 1146T^9 + 3183T^8 - 2697T^7 + 1867T^6 - 2241T^5 + 1728T^4 - 652T^3 + 135T^2 - 15*T + 1
$79$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + 6T^21 - 15T^20 + 38T^18 - 48T^16 - 181T^15 + 738T^14 - 1320T^13 + 1131T^12 - 1293T^11 + 2454T^10 - 2239T^9 + 1851T^8 - 2589T^7 + 1611T^6 - 465T^5 + 1020T^4 + 146T^3 + 171T^2 - 27*T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} - 14 T^{21} + \cdots + 1 $ T^24 - 14T^21 + 99T^18 - 258T^15 + 594T^12 + 154T^9 + 206T^6 + 28*T^3 + 1
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1767 = 3 \cdot 19 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1767.eh (of order \(90\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{90}^{37} q^{3} - \zeta_{90}^{26} q^{4} + ( - \zeta_{90}^{38} - \zeta_{90}^{34}) q^{7} - \zeta_{90}^{29} q^{9} + \zeta_{90}^{18} q^{12} + ( - \zeta_{90}^{28} - \zeta_{90}^{6}) q^{13} - \zeta_{90}^{7} q^{16} + \cdots + ( - \zeta_{90}^{37} + \zeta_{90}^{25}) q^{97} +O(q^{100}) \) q + z^37 * q^3 - z^26 * q^4 + (-z^38 - z^34) * q^7 - z^29 * q^9 + z^18 * q^12 + (-z^28 - z^6) * q^13 - z^7 * q^16 + z^24 * q^19 + (z^30 + z^26) * q^21 - z^5 * q^25 + z^21 * q^27 + (-z^19 - z^15) * q^28 - z^2 * q^31 - z^10 * q^36 + (z^38 + z^22) * q^37 + (-z^43 + z^20) * q^39 + (-z^20 + z^6) * q^43 - z^44 * q^48 + (-z^31 - z^27 - z^23) * q^49 + (z^32 - z^9) * q^52 - z^16 * q^57 + (z^16 + z^9) * q^61 + (-z^22 - z^18) * q^63 + z^33 * q^64 + (-z^36 + z^4) * q^67 + (-z^33 + z^29) * q^73 - z^42 * q^75 + z^5 * q^76 + (-z^35 - z^3) * q^79 - z^13 * q^81 + (z^11 + z^7) * q^84 + (z^44 + z^40 - z^21 - z^17) * q^91 - z^39 * q^93 + (-z^37 + z^25) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{12} - 3 q^{13} + 3 q^{19} - 12 q^{21} - 3 q^{27} - 12 q^{28} + 3 q^{43} - 6 q^{49} - 6 q^{52} + 6 q^{61} + 6 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 3 q^{75} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) \) 24 * q - 6 * q^12 - 3 * q^13 + 3 * q^19 - 12 * q^21 - 3 * q^27 - 12 * q^28 + 3 * q^43 - 6 * q^49 - 6 * q^52 + 6 * q^61 + 6 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 3 * q^75 + 3 * q^79 + 3 * q^91 + 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1767.1.eh.a

Level	$1767$
Weight	$1$
Character orbit	1767.eh
Analytic conductor	$0.882$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{90}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.881847877323\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{45})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{90}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{90} - \cdots)\)

\(n\)	\(590\)	\(685\)	\(838\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{90}^{24}\)	\(\zeta_{90}^{35}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 86.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
589.ce	even	90	1	inner
1767.eh	odd	90	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} \) T^24
$3$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$5$	\( T^{24} \) T^24
$7$	\( T^{24} + 6 T^{22} + \cdots + 1 \) T^24 + 6T^22 + 2T^21 + 27T^20 + 18T^19 + 111T^18 + 108T^17 + 456T^16 + 404T^15 + 1062T^14 + 1230T^13 + 2592T^12 + 3204T^11 + 5589T^10 + 6560T^9 + 8397T^8 + 7020T^7 + 5315T^6 + 2556T^5 + 1167T^4 + 294T^3 + 162T^2 - 21T + 1
$11$	\( T^{24} \) T^24
$13$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 3T^22 - T^21 - 30T^20 - 75T^19 - 72T^18 - 15T^17 + 222T^16 + 446T^15 + 363T^14 + 525T^13 + 516T^12 + 498T^11 + 579T^10 - 241T^9 + 1401T^8 + 2064T^7 + 1091T^6 + 330T^5 + 900T^4 + 444T^3 + 156T^2 - 18T + 1
$17$	\( T^{24} \) T^24
$19$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{3} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^3
$23$	\( T^{24} \) T^24
$29$	\( T^{24} \) T^24
$31$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$37$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 - 2T^21 + 90T^20 - 21T^19 + 425T^18 - 135T^17 + 1476T^16 - 524T^15 + 3618T^14 - 1419T^13 + 6625T^12 - 2556T^11 + 8322T^10 - 3139T^9 + 7560T^8 - 2664T^7 + 4193T^6 - 1377T^5 + 1563T^4 - 370T^3 + 180T^2 + 12T + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 - 2T^21 + 6T^20 - 21T^19 + 51T^18 - 54T^17 + 168T^16 - 510T^15 + 834T^14 - 762T^13 + 234T^12 - 702T^11 + 2535T^10 - 3311T^9 + 3948T^8 - 1812T^7 + 512T^6 + 384T^5 + 243T^4 + 298T^3 + 105T^2 + 15*T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} - 6 T^{23} + \cdots + 1 \) T^24 - 6T^23 + 21T^22 - 57T^21 + 135T^20 - 291T^19 + 541T^18 - 876T^17 + 1272T^16 - 1659T^15 + 1761T^14 - 696T^13 - 1720T^12 + 3921T^11 - 3483T^10 - 120T^9 + 2640T^8 - 1872T^7 + 392T^6 - 63T^5 + 462T^4 + 237T^3 + 3T^2 - 3*T + 1
$67$	\( T^{24} - 6 T^{23} + \cdots + 1 \) T^24 - 6T^23 + 21T^22 - 57T^21 + 135T^20 - 291T^19 + 541T^18 - 876T^17 + 1272T^16 - 1659T^15 + 1761T^14 - 696T^13 - 1720T^12 + 3921T^11 - 3483T^10 - 120T^9 + 2640T^8 - 1872T^7 + 392T^6 - 63T^5 + 462T^4 + 237T^3 + 3T^2 - 3*T + 1
$71$	\( T^{24} \) T^24
$73$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + 3T^21 - 24T^20 + 39T^19 + 6T^18 - 99T^17 + 183T^16 - 185T^15 + 129T^14 - 492T^13 + 994T^12 - 1872T^11 + 2325T^10 - 1146T^9 + 3183T^8 - 2697T^7 + 1867T^6 - 2241T^5 + 1728T^4 - 652T^3 + 135T^2 - 15*T + 1
$79$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + 6T^21 - 15T^20 + 38T^18 - 48T^16 - 181T^15 + 738T^14 - 1320T^13 + 1131T^12 - 1293T^11 + 2454T^10 - 2239T^9 + 1851T^8 - 2589T^7 + 1611T^6 - 465T^5 + 1020T^4 + 146T^3 + 171T^2 - 27*T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} - 14 T^{21} + \cdots + 1 \) T^24 - 14T^21 + 99T^18 - 258T^15 + 594T^12 + 154T^9 + 206T^6 + 28*T^3 + 1
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